#6440853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439035
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,21)(12,15)(13,18)(14,19)(16,20)(17,24)(22,25)(23,37)(26,27)(28,31)(29,33)(30,35)(32,36)(34,40)(38,39);(2,5)(6,8)(9,11,15,24,16,26,14,25)(10,17,19,21,20,22,12,27)(13,29,31,40,32,39,30,23)(18,34,35,33,36,37,28,38);(1,3,4,7)(2,6,5,8)(9,12,15,10,16,19,14,20)(11,22,24,27,26,17,25,21)(13,28,31,18,32,35,30,36)(23,38,29,34,40,33,39,37);(9,13)(10,18)(11,23)(12,28)(14,30)(15,31)(16,32)(17,33)(19,35)(20,36)(21,37)(22,38)(24,29)(25,39)(26,40)(27,34);(9,14,16,15)(10,19,20,12)(11,24,26,25)(13,30,32,31)(17,27,22,21)(18,35,36,28)(23,29,40,39)(33,34,38,37)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348049);